Mirror modifier can not be used in edit mode
The person who made the video is simply saying 'apply' instead of 'add'. They do not 'apply' a mirror modifier, they just 'add' one. At no point, at least around the time you mentioned, do they actually press the 'apply' button on the modifier.

If you were to apply the mirror modifier you would no longer have a mirroring effect on edits you make to the object. You only get mirroring for as long as the modifier is added to the object and not applied. When you are finished you can press 'apply' and the mirrored side of the mesh will now be editable like any other piece of geometry. Seeing as at that time in the video he hadn't made much geometry it wouldn't have made sense to apply it.

So, to summarise, to use the mirror modifier you just have to add it to the object by choosing it from the list of modifiers.